Very often we refer to this process as hand forging, although that is not exactly correct. As can be seen from the video, the axe is made with an automated hammer and a set of dies through which the axe head is progressively taken.
There is an even more automated method out there which uses a drop hammer. The main difference that I have seen is that a drop hammer requires less hits. It usually takes three to five hits with the drop hammer to produce the same result, again using a set of dies. Not all companies adopted the drop hammer method because of the high cost of the equipment.
The part that I found very interesting is that the video does not show the use of a lathe in making the handle. I have not looked at a Wetterlings handle closely, but I imagine one is used at some point.
To see an axe actually being hand forged, take a look at this video.